How do I use the algebraic method to balance the equation CaCO₃ + H₃PO₄ → Ca₃(PO₄)₂ + H₂CO₃?

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How do I use the algebraic method to balance the equation CaCO₃ + H₃PO₄ → Ca₃(PO₄)₂ + H₂CO₃?

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Answer 1 · 2014-09-16T22:00:19

The balanced equation is ${3CaCO}_{3} + {2H}_{3} {PO}_{4} \to {Ca}_{3} {({PO}_{4})}_{2} + {3H}_{2} {CO}_{3}$ $"3CaCO"_3 + "2H"_3"PO"_4 → "Ca"_3("PO"_4)_2 + "3H"_2"CO"_3$

Explanation:

You write the equation using letter variables for the coefficients.

$a {CaCO}_{3} + b H_{3} {PO}_{4} \to c {Ca}_{3} {({PO}_{4})}_{2} + d H_{2} {CO}_{3}$ $a"CaCO"_3 + b"H"_3"PO"_4 → c"Ca"_3("PO"_4)_2 + d"H"_2"CO"_3$

Then you set up a series of simultaneous equations, one for each element.

$Ca : a = 3 c$ $"Ca": a = 3c$
$C : a = d$ $"C": a = d$
$O : 3 a + 4 b = 8 c + 3 d$ $"O": 3a + 4b = 8c +3d$
$H : 3 b = 2 d$ $"H": 3b = 2d$
$P : b = 2 c$ $"P": b = 2c$

Now you solve the five simultaneous equations.

It looks almost impossible, but we know that the coefficients must be integers.

Let's set $c = 1$ $c = 1$

Then $a = 3$ $a = 3$ and

$d = a = 3$ $d = a = 3$

$b = 2 c = 2$ $b = 2c = 2$

So $a = 3$ $a = 3$ ; $b = 2$ $b = 2$ ; $c = 1$ $c = 1$ ; $d = 3$ $d = 3$

The balanced equation is

${3CaCO}_{3} + {2H}_{3} {PO}_{4} \to {Ca}_{3} {({PO}_{4})}_{2} + {3H}_{2} {CO}_{3}$ $"3CaCO"_3 + "2H"_3"PO"_4 → "Ca"_3("PO"_4)_2 + "3H"_2"CO"_3$

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How do I use the algebraic method to balance the equation CaCO₃ + H₃PO₄ → Ca₃(PO₄)₂ + H₂CO₃?

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